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On \(n\)-circled \(\mathcal{H}^ \infty\)-domains of holomorphy. (English) Zbl 0884.32011

A domain \(G\subset\mathbb{C}^n\) is called \(n\)-circled if for every \((z_2,\dots,z_n)\in G\) and any \((\theta_2,\dots,\theta_n)\in\mathbb{R}^n(e^{i\theta_1}z_1,\dots,e^{i\theta_n}z_n) \in G\). Let \({\mathcal F}\) be some subspace in \(H^\infty(G)\), for example \({\mathcal F}=A^k(G)\). \(G\) is said to be an \({\mathcal F}\)-domain of holomorphy if for any pair of domains \(G_0\), \(\widetilde G\subset\mathbb{C}^n\) with \(\emptyset\neq G_0\subset\widetilde G\cap G\), \(\widetilde G\not\subset G\), there exists a function \(f\in{\mathcal F}\) such that \(f|G_0\) is not the restriction of a function \(\widetilde f\in{\mathcal O}(\widetilde G)\).
The purpose of this paper is to present various characterizations of \(n\)-circled domains of holomorphy with respect to some distinguished subspaces of \(H^\infty(G)\).

MSC:

32D05 Domains of holomorphy
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